A Generalization of Fermat's Principle for Classical and Quantum Systems
AA Generalization of Fermat’s Principle for Classical and Quantum Systems
Tarek A. Elsayed ∗ Institute of Theoretical Physics, University of Heidelberg, Philosophenweg 19, 69120 Heidelberg, Germany
The analogy between dynamics and optics had a great influence on the development of the foun-dations of classical and quantum mechanics. We take this analogy one step further and investigatethe validity of Fermat’s principle in many-dimensional spaces describing dynamical systems (i.e.,the quantum Hilbert space and the classical phase and configuration space). We propose that if thenotion of a metric distance is well defined in that space and the velocity of the representative pointof the system is an invariant of motion, then a generalized version of Fermat’s principle will hold.We substantiate this conjecture for time-independent quantum systems and for a classical systemconsisting of coupled harmonic oscillators. An exception to this principle is the configuration spaceof a charged particle in a constant magnetic field; in this case the principle is valid in a framerotating by half the Larmor frequency, not the stationary lab frame.
1. INTRODUCTION
An important lesson that has been emphasizedthroughout the history of physics is that illuminating newaspects of the interwoven connections between geometryand physics leads to paradigm shifts in physics. Typi-cally, novel geometric considerations of physical quanti-ties lead to new variational principles which assign thenatural evolution of physical systems with an extremumof some functional or a geodesic curve in some hyper-space. The oldest of these variational principles is theFermat principle of least time, which became a funda-mental principle in geometric optics. The principle wasintroduced by Fermat, who also called it the principle ofnatural economy [1], and it states that light rays travel ina general medium along the path that minimizes the timetaken to travel between the initial and final destinations.The concept of natural economy inspired Maupertuis tointroduce the principle of least action in analytical me-chanics, which later evolved through the work of Euler,Lagrange, Hamilton, and Jacobi to become a fundamen-tal concept in classical mechanics. By 1887, it had be-come clear that the least action is a universal concept inphysics when Helmholtz expanded its domain of valid-ity by applying it to two regimes beyond the standardproblems of classical mechanics, namely, thermodynam-ics and electrodynamics [2]. Since then, the pursuit ofnew variational principles in physics has not relented [3].The mathematical formulation of Fermat’s principlestates that the time functional T , defined as T = (cid:90) dsν ( s ) , (1)where ν ( s ) is the speed of light and ds is the distanceelement along the light trajectory, is minimized [4]. Inother words, if T is computed along all possible trajec-tories between fixed initial and final positions, T willalways be minimum along the actual path traveled by ∗ Electronic address: [email protected] the light rays (the physical path). The modern versionof Fermat’s principle is written in terms of the index ofrefraction n ( s ) = cν ( s ) , where c is the speed of light in freespace, and states that the optical path length (cid:82) n ( s ) ds isa minimum. In that sense, Fermat’s principle is the opti-cal analog of Jacobi’s principle of least action [5], whichstates that for a conservative classical system at energy E , with potential function V between its constituent par-ticles, the action functional I = (cid:90) (cid:112) E − V ( s ) ds (2)is an extremum.The remarkable property of this action that distin-guishes it from other variational principles in analyti-cal mechanics, i.e., Hamilton and Lagrange’s variationalprinciples, is that it represents a purely geometric quan-tity. This quantity is computed along different trajecto-ries in the configuration space between fixed points with-out referring to any time evolution. Therefore, Eq. (2)can be used to define a new Riemannian space, whosemetric ds (cid:48) = (cid:112) E − V ( s ) ds , where the natural evolutionof the representative point of the system is along geodesiccurves.In this work, we set out to seek how far the analogybetween dynamics and optics applies as far as the Fermatprinciple is concerned. In particular, we investigate thevalidity of Fermat’s principle for a generic many-bodyclassical and quantum system, and pose the followingquestion: If the state of a conservative dynamical systemis represented in some metric space S by a point, andthe velocity field ν ( s ) is computed everywhere in S fromthe equations of motion using the proper metric of thatspace, will the motion of this point be along a path thatextremizes the time functional T ?
2. THE GENERALIZED FERMAT PRINCIPLE
We answer the question posed above by proposing the generalized fermat principle (GFP): Whenever the speedof the representative point of a conservative dynamical a r X i v : . [ qu a n t - ph ] S e p system, ν ( s ), is an integral of motion in a metric space S ,the path followed during the dynamical evolution of thatsystem in S between fixed initial and final states makesthe time functional T stationary against small variationsof the path. In contrast to light rays, where T is an ex-tremum even when the speed of light is not constant (i.e.,in an inhomogeneous medium), this conjecture considersonly the case when ν ( s ) is invariant during the time evo-lution. A corollary that follows from this conjecture isthat the length of the physical path (cid:82) ds is stationary(e.g., the path can be a geodesic) on the sub-manifoldof a given value of ν ( s ) embedded in S when the abovecondition is fulfilled.Mathematically speaking, the GFP states that if ν ( s )is a constant of motion along the physical path (not nec-essarily in the whole space), then among all possible tra-jectories between initial and final states, only those whichmake T invariant under an infinitesimal variation of thepath, i.e., δ T = 0 , (3)are possible candidates for the dynamical evolution. Thevalue of T corresponding to the physical path is not nec-essarily the global minimum between all paths connectingthe initial and final states. We emphasize here that weare not aiming to derive the equations of motion from thetime action, because we have to use them to find ν ( s ) inthe first place. We rather propose that they necessarilylead to a stationary time action when ν ( s ) is an integralof motion. Unlike the original Fermat principle, not everypair of states are connected by a physical path. Rather,the principle proposed here gives a geometrically appeal-ing argument to explain why the evolution of the systemfollowed a certain trajectory between a given pair of ini-tial and final states which we know a priori are connectedby some physical path.We investigate the validity of this conjecture by consid-ering three cases: (i) The evolution of quantum systemsin the projective Hilbert space P , where wavefunctionsare defined up to an overall phase factor. (ii) The evo-lution of a system of coupled harmonic oscillators in thephase space consisting of coordinates and momenta andequipped with an Euclidean metric. (iii) The motion of acharged particle in a constant magnetic field, which turnsout to be an exception. Similar to the Jacobi’s principle,the principle proposed here represents a geometric vari-ational principle in the phase and Hilbert space. In bothcases, the velocity field ν ( s ) is defined completely by theHamiltonian of the problem, and is obtained from theequations of motion of the system that will drive its evo-lution along the physical path (i.e., Schr¨odinger equationin quantum systems and Hamilton’s equations of motionin classical systems). A. Generalized Fermat Principle in Hilbert Space
The development of the concept of geometric phase inquantum mechanics triggered the interest of many physi-cists to look for more connections between quantum me-chanics and geometry [6]. Anandan and Aharonov inves-tigated the nature of the geometry of quantum evolutionin the projective Hilbert space P through a series of pa-pers in the late 80s [7–9]. They have shown [8] that thespeed of quantum evolution in P is related to the energyuncertainty ∆ E = (cid:16)(cid:10) H (cid:11) − (cid:104) H (cid:105) (cid:17) / via ds = ∆ E dt/ (cid:126) , (4)where ds is the infinitesimal distance in P given by theFubini-Study (FS) metric ds = (cid:104) δψ | δψ (cid:105)(cid:104) ψ | ψ (cid:105) − |(cid:104) δψ | ψ (cid:105)| (cid:104) ψ | ψ (cid:105) . Onthe unit sphere, ds = (cid:104) δψ | − ˆ P | δψ (cid:105) , where ˆ P is the pro-jection operator | ψ (cid:105)(cid:104) ψ | . The trajectory traversed by a rayin P under unitary evolution is generally not a geodesic,i.e., δ (cid:82) ds (cid:54) = 0. This can be easily conceived by consider-ing a system composed of a single quantum spin-1/2. Inthis case, P is simply the Bloch sphere and the precessionmotion of the spin on Bloch sphere off the equator is nota geodesic.Several attempts [10, 11] have been made to find newformulations where the quantum evolution is a geodesicflow. Noticing that the speed of quantum evolution∆ E/ (cid:126) is invariant for time-independent Hamiltonians,the simple answer to this problem suggested by thepresent paper is to consider T = (cid:82) ds ∆ E as a geodesicquantity, i.e., Fermat’s principle in Hilbert space . Thisissue should be distinguished from the quantum brachis-tochrone problem [12], where the Hamiltonian that leadsto optimal time evolution between an initial and fi-nal state is sought. The above proposition, however,states that the unitary evolution generated by any time-independent Hamiltonian is optimal, with respect to allother possible trajectories connecting the initial and finalstates (Fig. 1-a).To show that T is stationary along the physical paththrough P , let us parameterize the evolution along anypath connecting the initial and final states | ψ i (cid:105) and | ψ f (cid:105) by some arbitrary parameter τ . We can write Eq. (1) as T = (cid:90) | ψ f (cid:105)| ψ i (cid:105) dτ (cid:104) ˙ ψ | − ˆ P | ˙ ψ (cid:105) (cid:16) (cid:104) ψ | H | ψ (cid:105) − (cid:104) ψ | H | ψ (cid:105) (cid:17) , (5)where | ˙ ψ (cid:105) = | δψ (cid:105) δτ . Taking the variational derivative of T with respect to (cid:104) δψ | subject to the constraints of normal-ization and fixed initial and final states, we arrive at theEuler-Lagrange (EL) equation, δL (cid:104) δψ | − ddτ δL (cid:104) δ ˙ ψ | = 0 , (6)where L is the integrand in Eq. (5) added to the Lagrangemultiplier term λ ( τ )( (cid:104) ψ | ψ (cid:105) − | ψ (cid:105) . Calling the numer-ator and denominator in Eq. (5), A and B respectively,Eq. (6) reads (cid:20) − AB (cid:104) ˙ ψ | ψ (cid:105)| ˙ ψ (cid:105) − A B (cid:0) H | ψ (cid:105) − (cid:104) H (cid:105) H | ψ (cid:105) (cid:1)(cid:21) − AB (cid:104) | ¨ ψ (cid:105) − (cid:104) ψ | ˙ ψ (cid:105)| ˙ ψ (cid:105) − (cid:16) (cid:104) ˙ ψ | ˙ ψ (cid:105) + (cid:104) ψ | ¨ ψ (cid:105) (cid:17) | ψ (cid:105) (cid:105) − (cid:16) | ˙ ψ (cid:105) − (cid:104) ψ | ˙ ψ (cid:105)| ψ (cid:105) (cid:17) ∗ (cid:20) − AB (cid:16) (cid:104) ψH | ˙ ψ (cid:105) + (cid:104) ˙ ψ | H | ψ (cid:105) − (cid:104) H (cid:105) (cid:104) (cid:104) ˙ ψ | H | ψ (cid:105) + (cid:104) ψ | H | ˙ ψ (cid:105) (cid:105)(cid:17) − A B (cid:16) (cid:104) ¨ ψ | ˙ ψ (cid:105) − (cid:104) ψ | ˙ ψ (cid:105) (cid:16) (cid:104) ¨ ψ | ψ (cid:105) + (cid:104) ˙ ψ | ˙ ψ (cid:105) (cid:17) + c.c (cid:17)(cid:21) + λ ( τ ) | ψ (cid:105) = 0 . (7) Although Eq. (7) is a highly nonlinear equation, it is easyto verify that the Schr¨odinger equation | ˙ ψ (cid:105) = ± iH | ψ (cid:105) satisfies this equation with a vanishing Lagrange multi-plier, and therefore makes the time functional stationarywhen τ equals the real time t . The sign ambiguity canbe considered a reminiscence of the non-unique mappingbetween τ and t . In cases where quantum ergodicity ap-plies, i.e., when “all states within a given energy rangecan be reached from all other states within the range”[13], the opposite sign can be related to the other routeto reach | ψ f (cid:105) starting from | ψ i (cid:105) , i.e., backward in time.The above discussion provokes several interesting is-sues. First, it is intriguing to explore whether there isa nonlinear Schr¨odinger equation that would satisfy Eq.(7) with a vanishing Lagrange multiplier and fulfill theFermat principle as a possible extension to quantum me-chanics. It is unlikely that such an equation exists sinceany equation that satisfies Eq. (7) and keeps the normof | ψ (cid:105) conserved should also satisfy (cid:104) ˙ ψ | ˙ ψ (cid:105)|(cid:104) ψ | ˙ ψ (cid:105)| = (cid:10) ψ | H | ψ (cid:11) (cid:104) ψ | H | ψ (cid:105) . (8)This equation results after taking the inner product ofEq. (7) with (cid:104) ψ | and making use of the normaliza-tion condition of the wavefunction. Second, it has tobe emphasized that Eq. (6) is satisfied only for time-independent Hamiltonians. A very interesting problem ishow to generalize this concept to time-dependent Hamil-tonians as we shall do in the classical domain below. Fi-nally, we expect the Fermat principle to be equally validfor the unitary evolution of a density matrix with theFS metric replaced by the Hilbert-Schmidt metric. Itwould be interesting, though, to investigate whether thenon-unitary evolution of the density matrix of an openquantum system described by a master equation follows aFermat principle. We present no further details on theseissues in the present paper. B. Generalized Fermat Principle in Phase Space
In optics, the Euler-Lagrange equation for the func-tional c (cid:82) n ( s ) ds reduces to the ray equation (also called the eikonal equation) [14] dds ( n ( s ) ˆt ) − ∇ n ( s ) = 0 , (9)where ˆt is a unit tangent vector defined in terms of theposition vector r as ˆt = d(cid:126) r /ds (the length of the path s plays the role of time in this derivation). When n ( s ) isconstant along the path (i.e., independent of s ), Eq. (9)can be rewritten as dds ( ˆt ) = − ∇ ν ( s ) ν ( s ) . (10)The left hand side of this equation represents a curvaturevector (cid:126)κ whose magnitude equals the curvature of thepath and direction is orthogonal to the direction of mo-tion. Therefore, for the case of an Euclidean space S , theGFP is equivalent to stating that (cid:126)κ equals −∇ ν ( s ) /ν ( s )for a dynamical system that has invariable speed ν ( s )along its evolution in S . On the other hand, since (cid:126)κ isthe acceleration vector of the representative point of thesystem, we can regard the RHS of Eq. (10) as a force thatdrives its evolution. The potential function responsiblefor this force is log( ν ( s )).We now consider a generic conservative classical sys-tem composed of N particles described by a set of gener-alized coordinates { q i , p i } that have the same units andHamiltonian H (Fig. 1-b). Let the distance elementin the phase space be described by the Euclidean met-ric ds = (cid:80) Ni =1 dq i + dp i . The speed ν ( s ) along thephysical path generated by the Hamiltonian flow equals (cid:114)(cid:80) Ni =1 (cid:16) ∂H∂q i (cid:17) + (cid:16) ∂H∂p i (cid:17) [13]. We therefore express thetime functional T as T = (cid:90) (cid:126) x f (cid:126) x i ds (cid:114)(cid:80) i (cid:16) ∂H∂x i (cid:17) , (11)where x i denotes any of the generalized coordinates q i , p i treated on an equal footing and (cid:126) x denotes { x , x , ...x N } .If we parameterize any arbitrary trajectory connecting (cid:126) x i and (cid:126) x f by τ , then the EL equation which satisfies the | ψ (cid:105) | ψ (cid:105)| ψ N (cid:105)| ψ i (cid:105) | ψ f (cid:105) q p p N ( q i , p i ) ( q f , p f ) q q q N q i q f (a) (b) (c) FIG. 1: Natural evolution of a physical system from an initial to a final state (thick) on the unit sphere in the projective Hilbertspace (a), in the classical phase space (b) and in the classical configuration space (c) together with fictitious paths (dashed)connecting the same states. The time functional T is stationary when the speed of evolution defined by the Hamiltonian isconstant along the physical path. This condition is satisfied naturally for time independent Hamiltonians in case (a) and onlyin special cases of (b) and (c) (see text). variational principle δ T = 0 is δLδx − ddτ δLδ ˙ x = 0 , (12)where L = F G , F = (cid:114)(cid:80) i (cid:16) ∂H∂xi (cid:17) , G = (cid:113)(cid:80) i ˙ x i and x denotes any of the 2 N coordinates. The explicit form ofEq. (12) for coordinate x i is G ∂F∂x i − G (cid:16) ¨ x i F + ˙ x i (cid:80) j ∂F∂x j ˙ x j (cid:17) − ˙ x i F (cid:80) j ˙ x j ¨ x j G = 0 . (13)This equation is not generally satisfied for an arbitrarychoice of the generalized coordinates q i , p i . The only sys-tem known to the author for which a metric space isdefined and ν ( s ) is a constant of motion is the harmonicoscillator. It is easy to verify numerically that Eq. (13) issatisfied for a system of N coupled harmonic oscillatorswith equal masses and coupling constants with Hamilto-nian H = (cid:80) i p i m + (cid:80) i An obvious example where GFP is valid in configu-ration space is the free motion of a single particle con-strained to a curved surface following a geodesic trajec-tory. Similarly, for a system of free particles, GFP holdsin the configuration space { q i } (Fig. 1-c), while for a sys-tem of interacting particles it holds in the Riemannianmanifold defined by the metric ds (cid:48) introduced in the in-troduction. Another example is the bound states of thetwo-body Kepler problem. The 3D motion of the reducedone-body problem can be mapped onto the motion of afree particle on the inner surface of an S sphere embed-ded in a 4D space [16, 17]. In the rest of this section,we discuss the applicability of Fermat’s principle in theconfiguration space of another system, namely a charge q with mass m moving in a constant uniform magneticfield B .The dynamics of the charge controlled by the LorentzForce m d v dt = q v × B guarantees that the magnitude ofthe charge velocity is constant and GFP indicates thatthe time functional should be stationary in the config-uration space of the charge. However, in this case, astationary time functional T indicates that the length ofthe path of the charge in space is an extremum, which weknow is not the case; the circular path of the charge in aconstant magnetic field is not an extremal path. There-fore, we consider this case a clear counter-example to theGFP if we do not restrict it to velocity-independent po-tentials. However, we note that even in this case, thereis another frame of reference where GFP is valid. Whatwe will show next is that in a frame rotating with fre-quency ω = − q/ m B the time functional is a stationaryquantity and GFP restores its validity.The time derivative of the charge position vector inthe rotating frame d r dτ is related to its time derivative inthe lab frame d r dt by d r dτ = d r dt − ω × r [18]. By workingin the Coulomb gauge, A = − r × B / 2, it can be easilyshown that d r dτ = d r dt + q A /m . The RHS of this equationis nothing but the canonical momentum P = m v + q A divided by the mass of the charge. Let us call the ve-locities in the rotating frame and the lab frame u and v respectively. The time functional in the rotating frameis then T = (cid:82) dsu . If we parameterize the path in termsof the lab frame time coordinate t , the time functionaltransforms into T = (cid:82) vdtu . Assuming the magnetic fieldin the z direction and the charge initially having a hor-izontal velocity, we find that T = (cid:82) √ ˙ x + ˙ y dt √ ( ˙ x − ωy ) +( ˙ y + ωx ) where ω = | ω | . The author has verified numerically thatthe EL equations are verified for this action and hence T is a stationary quantity in the rotating frame. 3. DISCUSSION Having demonstrated the validity of the Fermat’s prin-ciple for quantum Hilbert spaces and the phase space ofa system of harmonic oscillators, we need to emphasizethat the validity of Eq. 3 is not a trivial consequenceof the invariance of ν ( s ) along the physical path, but israther attributed to the underlying dynamics, or moreprecisely the differential equations governing those dy-namics. Not every functional of the form (cid:82) F ( s ) ds is anextremum along the path of constant F . On the oppositeside, a dynamical system can have an extremum action (cid:82) F ( s ) ds even when F ( s ) is not a constant of motion asin Eq. 2. Moreover, the validity of Eq. 3 in both Hilbertand phase spaces may not be related to the fact that thespeed of evolution is directly connected with the energyin both cases since we can easily check that the func-tional (cid:82) (cid:104) H (cid:105) ds in the first case and (cid:82) H ( P i , Q i ) ds in thesecond case are not stationary along the physical path.Satisfying Eq. 3 is not synonymous to the conservationof energy, and we did not have to impose the constraintof constant energy to verify it. Moreover, on the energymanifolds in Hilbert and phase spaces, there is a multi-tude of trajectories connecting the initial and final states,but not all of them satisfy δ T = 0.While it is true that one can find a variational princi-ple that describes the solution of almost every differentialequation, the conceptual significance of the GFP is thelink it makes with an established and a well known prin-ciple in optics. The principle advertised in this paper isobviously not as practical as the original Fermat’s princi-ple or the Lagrange principle of least action for examplesince we cannot easily solve Eqs. 13 and 7 and find thephysical path. However, it illustrates an interesting prop-erty of the evolution in Hilbert and phase spaces that webelieve is worthy to be highlighted.As a leap of faith based on the previous examples, weclaim that GFP will be universally true in any spacewhere the dynamical system evolves with a constantspeed. This implies that if the state of the system isprojected to a sub-manifold of the full space S , where ametric is defined and ν ( s ) of the projected state is an integral of motion, Fermat’s principle will hold in thatspace as well. An important exception of this general-ization is the motion of a charge in a magnetic field; inthis case we found a new frame of reference where GFPis valid in its configuration space.The ability to find the manifold or the transformationthat renders ν ( s ) an integral of motion relies on our abil-ity to find the integrals of motion of the given dynamicalsystem, not a trivial task in many cases. Therefore, thegeneralized Fermat principle is more likely to be relevantin integrable classical systems than in non-integrable sys-tems. We recall that the original Fermat’s principle is awave phenomenon. The minimization of the time func-tional T occurs along the path which leaves the phaseof the light wave stationary with respect to small varia-tions of the path (see [19] and references therein.) Thenotion of periodic phase variations is found naturally intwo types of dynamical systems: completely integrableclassical systems and quantum systems. The descriptionof the system in terms of the action-angle variables inthe first case and the eigenbasis of the Hamiltonian inthe second case offers an intuitive representation of thedynamics as a superposition of waves .On the other hand, even for a completely integrablesystem, it is not a trivial task to find the metric spacewhere ν ( s ) is a constant of motion and it is not even clearwhether such a space exists for every integrable system.It is therefore instructive to ask: if we abandon the notionof single trajectories in the classical domain and considera semiclassical system described by a quasi-probabilitydistribution in the phase space, can a generalized form ofFermat’s principle hold in this representation? It seemsplausible to expect interesting phenomena to occur in thiscase due to the quantum corrections to the classical Li-ouville’s equation or because quantum quasi-probabilitywavepackets can be thought of as superpositions of manyclassical trajectories that interfere with each other [20].We hope that this discussion will trigger more researchefforts into this direction. 4. CONCLUSION In conclusion, we have introduced a conjecture thatgeneralizes Fermat’s principle in Hilbert and phase spacesof quantum and classical systems respectively and illu-minates new aspects of Fermat’s surmise of natural econ-omy . The generalized Fermat principle provides a newgeometric variational principle satisfied naturally by theSchr¨odinger and Hamilton’s equations of motion in theproper space that has an associated metric distance whenthe speed of evolution is an integral of motion. This prin-ciple may have implications for the protein folding prob-lem, one of the major challenges of biological sciences to-day [21]. The mechanism followed by a protein to evolvefrom its unfolded structure to the native structure thathas a minimum free energy is not fully understood. Thepuzzle lies in the ultrashort time the protein takes to foldwith respect to the astronomical number of intermediateconformations. Although the folding protein is an opensystem, we may gain some insight into this problem bysearching for the proper space S where the generalized Fermat principle is fulfilled.The author would like to thank Chris Gray for his com-ments and Boris Fine for several fruitful discussions anduseful comments on the manuscript. [1] J.-L. Basdevant, Variational Principles in Physics (Springer, 2007).[2] H. von Helmholtz, Journal f¨ur die reine und angewandteMathematik (Crelle’s Journal) , 137 (1887).[3] C. G. Gray, G. Karl, and V. A. 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